Other Common Asymptotic Notations and Why Big O Trumps Them

Big Omega: $\Omega(.)$

Mathematically, a function $f(n)$ is in $\Omega(g(n))$ if there exists a real constant $c > 0$ and there exists $n_o > 0$ such that $f(n) \geq cg(n)$ for $n \geq n_o$. In other words, for sufficiently large values of $n$, $f(n)$ will grow at least as fast as $g(n)$.

Note: It is a common misconception that Big O characterizes the worst-case running time while Big Omega characterizes the best-case running time of an algorithm. There is no one-to-one relationship between any of the cases and the asymptotic notations.

The following graph shows an example of functions $f(n)$ and $g(n)$ that have a Big Omega relationship:

Note: Did you know that if $f(n) \in O(g(n))$, then $g(n) \in \Omega(f(n))$? Try proving it!

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Big Theta: $\Theta(.)$

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